PhD Defense: Valerio Proietti

Title:
K-theory, groups, and topological dynamics

Abstract:
This thesis studies the K-theory of groupoid C*-algebras and its applications to topological dynamics and index theory.

We introduce a homology theory for groupoids admitting an open ``computable'' subgroupoid. This is part of a work-in-progress project whose objective is computing the K-groups of C*-algebras associated to hyperbolic dynamics. The main tool is an induction-restriction adjunction in the setting of equivariant Kasparov categories, which also allows for a general formulation of the strong Baum-Connes conjecture for étale groupoids.

Paper A (joint work with Jens Kaad) focuses on the assembly map for principal bundles with fiber a countable discrete group. We derive Atiyah’s L^2-index theorem in the general context of flat C*-module bundles over compact Hausdorff spaces. Our approach does not rely on geometric K-homology but rather on a Chern character construction for Alexander-Spanier cohomology.

Paper B deals with the homology for Smale spaces defined by Putnam. We introduce a simplicial framework by which the various complexes attached to this theory can be understood as suitable ``symmetric'' Moore complexes. We prove they are all quasi-isomorphic and discuss a parallel with sheaf cohomology by computing the projective cover of a Smale space. 

Supervisor:
Prof. Ryszard Nest, MATH, University of Copenhagen

Assessment Committee:
Prof. (Chairman), Henrik Schlichtkrull, University of Copenhagen
Prof. Sergey Neshveyev, UIO, Norway
Prof, Michael Whittaker, University of Glasgow, UK